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I have seen conformal mappings from convex polygons to the unit circle, and I have seen mappings from the circle to non-convex polygons. I have not seen arbitrary simple non-convex polygons mapped to unit circles, which is where my interest lies.

Does such a mapping exist? In either case, what is an example, and what is a specific reference that I can follow up with?

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    $\begingroup$ Isn't this precisely what Riemann mapping theorem says? en.wikipedia.org/wiki/Riemann_mapping_theorem $\endgroup$ – xyzzyz Aug 14 '17 at 18:24
  • $\begingroup$ Certainly the inverse of such a mapping exists by RMT, so you should be able to compute it somehow $\endgroup$ – DaveNine Aug 15 '17 at 0:43
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As comments suggest, such a mapping exists by the Riemann Mapping Theorem. A constructive approach to determining the mapping (and its inverse) is given by the Schwarz-Christoffel formula. For reference, the book Schwarz-Christoffel Mapping by Driscoll and Trefethen is helpful.

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